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A math challenge exercise and challenge for those of you good at math

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Fear The Spear

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A friend of mine works at a local outdoor market and freshly-squeezes lemon to sell lemonade. She mixes the fresh lemon juice with a sugar-water mixture to create the drink. She wants to be able to tell the customers how much sugar is in each drink.

Since I am somewhat good at math, she asked me to do the calculation.
She basically wants to know how many ounces of sugar are in each 1 oz of water (actually the sugar-water mixture). Then she'll just convert that to grams and multiply based on the size of the drink.

Basically what she does, is she mixes 5 lbs of sugar with 4 gallons of water.
So after that's combined, how much sugar is in each OUNCE of the liquid ?

I came up with .16 of an ounce of sugar, per 1 oz of the liquid.
Or converting the sugar to grams, I get 4 grams of sugar, per each ONE OUNCE of the water mixture.

Let me know what you come up with it. If you come up with a different answer than I did, I'll go ahead and show how I came up with my total, and you do the same.

Note that this is BEFORE the lemon juice is added, so the liquid is simply the sugar-water mixture. I also realize that there is fruit sugar in lemons, so the total sugar in the drink when the lemon juice is added will actually be greater. But for the sake of simplicity, let's disregard that element.

ALSO NOTE : I just realized that if you add 5 lbs of sugar to 4 gallons of water, even though the sugar dissolves, the sheer volume of the dissolved sugar must actually increase the total liquid mixture amount to more than 4 gallons, does it not ?
And if so, then would it be more accurate to find out how much the liquid mixture increases by, and use that figure, for all subsequent calculations, as opposed to the original "4 Gallon Liquid Amount" ?

Follow me so far ?
 

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A friend of mine works at a local outdoor market and freshly-squeezes lemon to sell lemonade. She mixes the fresh lemon juice with a sugar-water mixture to create the drink. She wants to be able to tell the customers how much sugar is in each drink.

Since I am somewhat good at math, she asked me to do the calculation.
She basically wants to know how many ounces of sugar are in each 1 oz of water (actually the sugar-water mixture). Then she'll just convert that to grams and multiply based on the size of the drink.

Basically what she does, is she mixes 5 lbs of sugar with 4 gallons of water.
So after that's combined, how much sugar is in each OUNCE of the liquid ?

I came up with .16 of an ounce of sugar, per 1 oz of the liquid.
Or converting the sugar to grams, I get 4 grams of sugar, per each ONE OUNCE of the water mixture.

Let me know what you come up with it. If you come up with a different answer than I did, I'll go ahead and show how I came up with my total, and you do the same.

Note that this is BEFORE the lemon juice is added, so the liquid is simply the sugar-water mixture. I also realize that there is fruit sugar in lemons, so the total sugar in the drink when the lemon juice is added will actually be greater. But for the sake of simplicity, let's disregard that element.

ALSO NOTE : I just realized that if you add 5 lbs of sugar to 4 gallons of water, even though the sugar dissolves, the sheer volume of the dissolved sugar must actually increase the total liquid mixture amount to more than 4 gallons, does it not ?
And if so, then would it be more accurate to find out how much the liquid mixture increases by, and use that figure, for all subsequent calculations, as opposed to the original "4 Gallon Liquid Amount" ?

Follow me so far ?
Honestly, it's technically an illogical question to answer, and I'll tell you why. Since the sugar is in pounds, which is a measure of weight, and the water is in gallons, which is a measure of volume, you can not accurately find out the answer without breaking it down into parts per million, then sorting it out from there.

But, since sugar is water soluble, you could find the answer by finding out first how many fluid ounces 5 lbs of sugar would basically melt down into, then base the measurement of that volume combined with the volume of water.

All that being said, there is a pretty elaborate formula for this that I will gladly work out for you, then be back shortly with the answer.

edit: you are more or less pretty damn close to dead on. The exact amount is: 0.1560459447764 ounces of sugar per fluid ounce of water.

In other words, you did good. I would have thought more sugar would be "lost" in the dissolving process. Who knew.

As for your note on the end, which I admittedly just read now, you are correct, which is what I was solving for. It's all about converting solubility and displacement.
 
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Fear The Spear

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Honestly, it's technically an illogical question to answer, and I'll tell you why. Since the sugar is in pounds, which is a measure of weight, and the water is in gallons, which is a measure of volume, you can not accurately find out the answer without breaking it down into parts per million, then sorting it out from there.

But, since sugar is water soluble, you could find the answer by finding out first how many fluid ounces 5 lbs of sugar would basically melt down into, then base the measurement of that volume combined with the volume of water.
Yes, I actually foresaw the dilemma of matching up weight (pounds) versus volume (gallons). So I used this website's conversion. It actually specifically gives sugar as an example, and says :
"2 cups sugar weighs approximately 1 pound.
But for flour you have 4 cups per pound."

http://wiki.answers.com/Q/How_many_cups_are_in_a_pound

All that being said, there is a pretty elaborate formula for this that I will gladly work out for you, then be back shortly with the answer.

edit: you are more or less pretty damn close to dead on. The exact amount is: 0.1560459447764 ounces of sugar per fluid ounce of water.

In other words, you did good. I would have thought more sugar would be "lost" in the dissolving process. Who knew.
Actually, I rounded off my number. My full answer was .15625, which is not quite your figure, but actually a whole lot closer to your number, than even my "rounded-off figure" of .16
 
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